Figure 2

An example of truth table, state transition diagram, and transition probability matrix of a PBN. The truth table, the state transition diagram, and the transition probability matrix A of a PBN without perturbations consisting of three variables V = {x₁,x₂,x₃} and$\mathcal{F}=(F_1,F_2,F_3)$, where$F_1=\left\{f_1^{\left(1\right)}\right\}$,$F_2=\{f_1^{\left(2\right)},f_2^{\left(2\right)}\}$, and$F_3=\{f_1^{\left(3\right)},f_2^{\left(3\right)}\}$. Since there is one predictor function for node x₁ and two predictors for nodes x₂ and x₃, there are 1 · 2 · 2 = 4 realisations of the PBN given by four network transition functions${\mathit{f}}_1=(f_1^{\left(1\right)},f_1^{\left(2\right)},f_1^{\left(3\right)})$,${\mathit{f}}_2=(f_1^{\left(1\right)},f_1^{\left(2\right)},f_2^{\left(3\right)})$,${\mathit{f}}_3=(f_1^{\left(1\right)},f_2^{\left(2\right)},f_1^{\left(3\right)})$, and${\mathit{f}}_4=(f_1^{\left(1\right)},f_2^{\left(2\right)},f_2^{\left(3\right)})$ with associated probabilities c₁ = 0.12, c₂ = 0.18, c₃ = 0.28, and c₄ = 0.42, respectively. For example,$c_3=c_1^{\left(1\right)}·c_2^{\left(2\right)}·c_1^{\left(3\right)}=1·0.7·0.4=0.28$. The edges in the state transition diagram are labelled with the transition probabilities. As can be seen from the state transition diagram, the underlying Markov chain is irreducible and aperiodic, thus ergodic. The steady-state (limiting) distribution for the chosen c _i values, i = 1..4, is given by$[\frac{7}{1609},\frac{3640}{14481},\frac{49}{4827},\frac{716}{4827},\frac{175}{4827},\frac{238}{4827},\frac{2548}{14481},\frac{4696}{14481}]$ (the states are considered in the lexicographical order from 000 to 111).